##
**Twisted loop groups and their affine flag varieties. With an appendix by T. Haines and M. Rapoport.**
*(English)*
Zbl 1159.22010

This paper consists of the main article written by Pappas and Rapoport, and an Appendix on parahoric subgroups written by T. Haines and Rapoport.

We first review the main article. Let \(k\) be a perfect field, and let \(K=k((t))\) be the field of Laurent series over \(k\). The authors develop a theory of affine Grassmannians and, more generally, (partial) affine flag varieties for a reductive group \(G\) over \(k((t))\). This generalizes the well-known results in the case where the group is assumed to be defined over \(k\). The basic construction is very similar to the classical case. The affine flag varieties are constructed as fpqc-sheaf quotients of the loop group by Bruhat-Tits group schemes associated to parahoric subgroups; they are ind-schemes over \(k\). The main results are the following theorems:

Assume that \(k\) is algebraically closed. Then there is a natural bijection (“Kottwitz homomorphism”) from the set of connected components of the loop group (or any of the partial affine flag varieties) onto the set of coinvariants under the Galois group \(\text{Gal}(\overline{K}/K)\) of the algebraic fundamental group \(\pi_1(G)\).

Now suppose that \(G\) is semisimple and splits over a tamely ramified extension of \(K\), and that the order of the fundamental group of the derived group of \(G\) is prime to the characteristic of \(k\). Then the loop group and the partial affine flag varieties are reduced ind-schemes.

Finally, suppose that \(G\) splits over a tamely ramified extension of \(K\), and that the order of the fundamental group of the derived group of \(G\) is prime to the characteristic of \(k\). Then all Schubert varieties in a partial affine flag variety are normal and have rational singularities. In positive characteristic, they are all Frobenius split.

In the final section of their paper, the authors discuss how to apply their results to the local models of Shimura varieties associated to ramified unitary groups; cf. G. Pappas and M. Rapoport [Local models in the ramified case. III. Unitary groups, preprint math.AG/0702286]. These applications are based on a coherence conjecture by the authors, which concerns the dimensions of the spaces of global sections of the natural ample invertible sheaves on partial flag varieties attached to \(G\). This conjecture can be stated in purely combinatorial terms.

In the Appendix, which also is of independent interest, Haines and Rapoport prove a number of facts about parahoric subgroups and Iwahori-Weyl groups.

We first review the main article. Let \(k\) be a perfect field, and let \(K=k((t))\) be the field of Laurent series over \(k\). The authors develop a theory of affine Grassmannians and, more generally, (partial) affine flag varieties for a reductive group \(G\) over \(k((t))\). This generalizes the well-known results in the case where the group is assumed to be defined over \(k\). The basic construction is very similar to the classical case. The affine flag varieties are constructed as fpqc-sheaf quotients of the loop group by Bruhat-Tits group schemes associated to parahoric subgroups; they are ind-schemes over \(k\). The main results are the following theorems:

Assume that \(k\) is algebraically closed. Then there is a natural bijection (“Kottwitz homomorphism”) from the set of connected components of the loop group (or any of the partial affine flag varieties) onto the set of coinvariants under the Galois group \(\text{Gal}(\overline{K}/K)\) of the algebraic fundamental group \(\pi_1(G)\).

Now suppose that \(G\) is semisimple and splits over a tamely ramified extension of \(K\), and that the order of the fundamental group of the derived group of \(G\) is prime to the characteristic of \(k\). Then the loop group and the partial affine flag varieties are reduced ind-schemes.

Finally, suppose that \(G\) splits over a tamely ramified extension of \(K\), and that the order of the fundamental group of the derived group of \(G\) is prime to the characteristic of \(k\). Then all Schubert varieties in a partial affine flag variety are normal and have rational singularities. In positive characteristic, they are all Frobenius split.

In the final section of their paper, the authors discuss how to apply their results to the local models of Shimura varieties associated to ramified unitary groups; cf. G. Pappas and M. Rapoport [Local models in the ramified case. III. Unitary groups, preprint math.AG/0702286]. These applications are based on a coherence conjecture by the authors, which concerns the dimensions of the spaces of global sections of the natural ample invertible sheaves on partial flag varieties attached to \(G\). This conjecture can be stated in purely combinatorial terms.

In the Appendix, which also is of independent interest, Haines and Rapoport prove a number of facts about parahoric subgroups and Iwahori-Weyl groups.

Reviewer: Ulrich Görtz (Bonn)

### MSC:

22E67 | Loop groups and related constructions, group-theoretic treatment |

14G35 | Modular and Shimura varieties |

20G25 | Linear algebraic groups over local fields and their integers |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

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\textit{G. Pappas} and \textit{M. Rapoport}, Adv. Math. 219, No. 1, 118--198 (2008; Zbl 1159.22010)

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